# Northeastern REU

## Independent Research Experience for Undergraduates

## REU 2019

## Calendar

**May 6: First Official REU Meeting (Summer I)****May 13: Proposal Draft****May 28: REU Tea & Colloquium 3-4 PM, Lake 509****June 4: REU Tea & Group Presentations 3-4 PM, Nightingale 544****June 11: REU Tea & Colloquium 3-4 PM, Lake 509****June 21: Final Presentations, Lunch 12-1, Talks 1-4, Lake 509 (Summer I)****June 28: Final Paper Due (Summer I)****August 14: Final Presentations (Summer II)**

**Final Paper: **At least 5 pages. If two students, then paper is written together is at least 10 pages. Should include an introduction with motivation and guiding questions, definitions, examples, a main result and proof. No maximum page limit, but writing should be clear and concise.

**Final Presentation: **30 minutes (40 if group of 2). Should introduce audience to area of research, convince them it is interesting through examples, definitions and results. Time is too short to include everything so choose carefully what to present highlighting what you worked hardest on. Narrative and slides and/or boardwork should be clear.

## Groups

### Evolutionary Dynamics.

**Participants**: Hoyin Chu. **Mentor**: Jonier Antunes. Summer I.

**Paper: ***Introduction to Evolutionary Dynamics and Stochastic Calculus.*

### Knot Theory.

**Participants**: Noah Fleischmann, Kally Lyonnais. **Mentor**: Dmytro Matvieievsky. Summer I.

**Paper: ***Knots, Colors, and Polynomials .*

### Graph Theory/Chip Firing:

**Participants:** Kristin Timothy, Michael Wang. **Mentor: **Ian Dumais. Summer I.

Paper: *Sandpiles and Chip Firing.*

### Homotopical Algebra:

**Participants:** Karthik Boyareddygari **Mentor: **Oleksiy Sorokin. Summer I.

**Paper: ***Homotopical Approach to Tensor Products.*

### Algebraic Geometry:

**Participants:** Walker Miller-Breetz. **Mentor: **Anupam Kumar. Summer II.

Paper: *Cone Points on Algebraic Curves*.

## Future Years

The RTG group welcomes independent and motivated undergraduates to apply for our Research Experience for Undergraduates (REU). Undergraduate students work with PhD-student mentors for two months during the summer on a research topic, under the supervision of a faculty member. Candidates that are citizens, nationals, or permanent residents of the United States or its territories and possessions are eligible for a scholarship of $2,000 per month.

If you would like to participate in the future, we would welcome you to apply. The application deadline will likely be end of February. You can view the old application form here, though the application may vary.

Potential topics include: Derived categories and homological algebra, Birational geometry and moduli spaces, Hyperkähler manifolds, Moduli theory and enumerative invariants, Hodge theory, Mirror symmetry, Geometric representation theory, Representations of Lie groups and Lie algebras, Quantum groups, Symplectic and Poisson geometry, Geometric quantization, Quantum field theory and string theory, Combinatorial Geometries, Toric Varieties, Arrangements of algebraic varieties.

For more information, please contact MATH-REU@LISTSERV.NEU.EDU

Northeastern University particularly welcomes applications from minorities, women, and persons with disabilities.

## REU 2018

The inaugural year of the Northeastern REU had 9 undergraduate participants working in several different areas. The postdoc coordinator was Ivan Martino.

- Student:
**Alon Duvall**

Mentor: Brian Hepler

Title:* Abstract Regular Polytope*

Abstract: Introduction to Abstract Regular Polytopes through a combinatorial geometry approach. C-Groups, Coxeter Groups and 2^k-constructions are also studied.

- Student:
**Kevin Su**

Mentor: Jonier Antunes

Title: *Graph Limits and Extremal Combinatorics*

Abstract: "One of the problems in extremal combinatorics is asking how many edges a graph on n nodes may have while avoiding some speciﬁc subgraphs. For example, Mantel’s theorem says that the most edges a graph on n nodes can have while avoiding K_3 is n^2/4. These statements can also be represented in terms of inequalities of numbers of homomorphisms or in terms of homomorphism densities. Many of these inequalities may be proved using Cauchy-Schwarz (as an inequality on sums of squares) or pictorially due to a gluing algebra on the space of graphs. We summarize some of the graph algebra background involved here and look at how graph limit theory gives a way of proving results in extremal combinatorics."

- Student:
**Timothy Jackman**

Mentor: Whitney Drazen

Title: *Graph Theory*

Abstract: This is an introduction to spectral graph theory until the concepts of (quantum) state transfer.

- Student:
**Felipe Castellano-Macias**

Mentor: Alex Sorokin

Title: *Leavitt Path Algebras** *

Abstract. We describe diﬀerent properties of Leavitt path algebras of a graph over a ﬁeld, providing the appropriate background. In addition, we investigate similar results about Leavitt path algebras of a graph over a commutative unital ring. We will mainly explore the connection between the diagonalizability of matrices over a given Leavitt path algebra and the structure of its underlying graph, as well as the classiﬁcation of such algebras up to isomorphism.

- Name:
**Benjamin Bonenfant**

Mentor: Reuven Hodges

- Student:
**Walker Miller-Breetz**

Mentor: Rahul Singh

Title: Young Tableux

Abstract: After a short introduction on group theory and group actions, the work focuses on the symmetric group and its action on flagged spaces.

- Student:
**Christina Nguyen**

Mentor: Whitney Drazen

Title: *Non-backtracking Walks on Graphs*

Abstract: The main goal of the REU was to obtain a new proof of the non-backtracking version of Ploya’s Theorem for random walks. We began with a discussion of spectral graph theory before studying the backtracking version of Ploya’s Theorem.

- Student:
**Noah Lichtblau**

Mentor: Mikhail Mironov

Title: *On the Morphism of Schemes*

- Student:
**Zheying Yu**

Mentor: Celine Bonandrini

Title:* Topology*